Physics 221
Quizes and Examinations
Fall 2006
Porter Johnson
“Physics can only be learned by thinking, writing, and worrying.”
-David Atkinson and Porter Johnson (2002)
“There is no royal road to geometry.”
-Euclid
1. PHYS 221 - 005, QUIZ 1, 31 August 2006
A block of mass M = 5kg, initially at rest on a horizontal table, is attached
to a rigid support by a spring with spring constant k = 5000N/m. A bullet
of mass m = 5 grams and speed of v = 1000 meters per second strikes and
is imbedded in the block. Assume that the compression of the spring is
negligible until the bullet is imbedded, and determine the following:
• the speed of the block immediately after the collision.
• the maximum compression of the spring.
• the period of the resulting simple harmonic motion.
Solution:
During the collision mechanical energy is converted into heat, whereas momentum is conserved. Thus, if the final speed of bullet plus block is V ,
then
m v = (M + m) V
or
V = mv/(M + m) = (5/5005) · 1000 ≈ 1 m/s
If the maximum compression of the spring is d, then the kinetic energy of
block plus bullet just after the collision is equal to the compressional energy
at maximum compression:
1 2
1
kd =
(M + m)V 2
2
2
d 2 = (M + m)V 2 /k = (5.005)(1)2/(5000) ≈ 0.001
d ≈ 0.03 m
From Newton’s second law for the mass plus string, the displacement of the
spring, x(t), satisfies the differential equation
(M + m) ẍ = F = −kx
or
a=−
k
x = −ω2 x
M +m
so that ω2 = k/(M + m) = 5000/(5.005) ≈ 1000. Thus ω ≈ 32 rad/sec,
corresponding to a period T = 2π/ω ≈ 0.2 sec.
This is the basis for the ballistic pendulum, which occasionally appeared
in general physics laboratories several decades ago. It was also used by
forensic specialists to measure muzzle speeds. What do they do now?
2. PHYS 221 - 006, QUIZ 1, 01 September 2006
A loudspeaker diaphragm is oscillating in simple harmonic motion with a
frequency of f = 500 Hz and a maximum displacement of A = 1.0 mm.
Determine the following:
• the angular frequency of vibration.
• the maximum speed of the diaphragm.
• the maximum acceleration of the diaphragm.
Solution:
Simple harmonic motion:
ω
x
dx
v=
dt
dv
a=
dt
= 2π f = 1000 π ≈ 3100 rad/sec
= A sin ωt
= ωA cos ωt
= −ω2 A sin ωt
Thus
vmax = ωA ≈ 3100 · (0.001) = 3.1 m/s
amax = ω2 A ≈ (3100)2 · (0.001) ≈ 9800 m/s2
Note: The maximum acceleration corresponds to 1000 g, and the diaphragm
will surely be ripped to pieces. Why is vibration at high frequency more
damaging than at low frequency?
3. PHYS 221 - 003, QUIZ 1, 07 September 2006
The speed of a transverse wave on a string is vT = 200 meters per second,
when the string tension is T = 150 Newtons.
• Determine the mass per unit length µ of the string.
• What tension T ′ is required in the string to raise the wave speed to
v′ = 250 meters per second?
Solution:
The speed of transverse vibrations is v =
length is
p
T /µ, so that the mass per unit
µ = T /v2 = (150)/(200)2 = 3/800 ≈ 0.0038 kg/m
A
pstring of mass per unit length µ and tension T has transverse speed v =
T /µ.
p
′
The same string, with tension T ′ , has transverse vibration speed v = T ′ /µ.
Thus
r
′
v
T′
=
v
T
!2
′
v
250 2
′
T =T
= 150
≈ 230 N
v
200
Ergo · · · the greater the tension the higher the pitch. You can check it out on
a guitar – even a toy guitar.
4. PHYS 221 - 004, QUIZ 1, 12 September 2006
A string of length L = 1.5 meters has a mass of m = 4.0 grams. It is stretched
between fixed supports with a tension of T = 25 Newtons.
• Determine the speed of transverse vibrations of the string.
• What is the lowest resonant frequency of this string, in Hertz?
Solution:
The mass per unit length is
µ = m/L = (0.004 kg)/(1.5 m) ≈ 2.7 × 10−3 kg/m
Thus the velocity is
v=
s
T
=
µ
r
√
25
9200 ≈ 96 m/s
≈
2.7 × 10−3
For the lowest resonant mode, λ = 2L = 3 meters, and f = v/λ ≈ 96/3 =
32 Hz. This is an “elephant rumble”.
5. PHYS 221 - 005, QUIZ 2, 14 September 2006
A point source emits P = 50 watts of sound isotropically. A small microphone intercepts the sound in an area of Amike = 0.5 cm2 , a distance R = 200
meters from the source. Calculate the power Pmike intercepted by the microphone, in Watts, as well as the sound intensity Imike at the microphone, in
Watts per meter-squared and in deciBels.
Solution:
The isotropic source has intensity I = P/(4πR2 ) at a distance R, so that
Imike = 50/(4π 2002 ) ≈ 9.95 × 10−5W /m2 . The corresponding deciBel
reading is
β = 10 · log10
≈ 10 · log10
Imike
Ithreshold
9.95 × 10−5
10−12
= 10 · log10 (9.95 × 107) = 10 · (7.9097) ≈ 79 dB.
The sound is thus loud, but not painful. Recall that a hundred Watt amplifier requires 100W of electric power, but converts only a small amount of it
into acoustic energy.
The power into the microphone is
Pmike = Imike Amike = (9.95 × 10−5) · (5 × 10−4) ≈ 5 × 10−9 W
A few nanoWatts, thus. Be sure to stand close to the microphone when you
speak!
6. PHYS 221 - 006, QUIZ 2, 15 September 2006
A pipe L = 1.0 meters long and closed at one end is filled with an unknown
gas. The third lowest harmonic frequency is measured to be f = 600 Hz.
• What is the speed of sound v in the unknown gas?
• What is the fundamental frequency f0 for this pipe when it is filled
with the unknown gas?
Solution:
For the pipe of length L with one end closed the fundamental mode corresponds to a quarter wavelength; λ = L/4. (a node at closed end, and an
antinode at the open end). The resonant wavelengths are L = (2n + l)λ/4,
for n = 0, 1, 2, · · ·.
The third wavelength, n = 2, gives l = 5λ2 /4, so that λ2 = 4L/5 = 0.8 m.
This mode has a frequency f2 = 600 Hz, so that the speed of sound inside
the pipe is v = λ f = 0.8 · 600 = 480 m/s.
For the fundamental mode
λo = 4L = 4 m
fo = v/λo = 480/4 = 120 Hz
The speed of sound on a gas depends upon the atom / molecule involved, as
well as the temperature. What gas is in this tube?
7. PHYS 221 - 003, QUIZ 2, 21 September 2006
Two equal positive charges, +Q, are placed on opposite sides of a square of
side a. Two equal negative charges, −q, are placed on the two other sides.
Determine the ratio Q/q, if the net force on each of the positive charges +Q
is zero.
Solution:
Put the postive charges +Q at locations (0, 0) and (a, a), with the negative
charges −q at locations (a, 0) and (0, a).
The forces on the charge at the √
origin come from the other three charges.
The charge +Q lies at distance 2a, so that the magnitude of the force ~F2
is kQ2 /(2a2). The two negative charges, which lie at distance a from that
charge, produce forces ~F1 and ~F3 , respectively. They are each of magnitude
kQq/a2. We separate these forces into components as indicated below:
Location Force
~1
(a, 0)
F
~2
(a, a)
F
(0, a)
x-component y-component
kQq
|
0
a2
|
~3
F
~Ftotal
|
|
−
kQ2
√
2 2a2
0
0
−
kQ2
√
2 2a2
kQq
a2
0
For the net force on the charge at the origin to be zero, we must have
kQq
kQ2
√
=
a2
2 2a2
√
or q = Q/(2 2).
The net electric field on the charge at (a, a) also vanishes, by symmetry.
Note: The forces on the negative charges do not vanish. They go flying
away from one another in a diagonal direction –√never to return! The net
electric force on the negative charge at (0, a) is ( 2 − 1) · kQ2/(2a2).
8. PHYS 221 - 004, QUIZ 2, 26 September 2006
Two positive charges, each of magnitude Q = 10−6 Coulombs are placed
symmetrically on opposite sides of an insulating wire, each a distance d = 1
meter from the wire along a line perpendicular to the wire. A negative
charge q = −10−6 Coulombs is imbedded on a bead that slides smoothly
along the wire. The mass of the bead is m = 0.001 kg.
• If the bead lies a (small) distance x to the right of the center line joining
the positive charges, determine the net force on the bead.
• Determine the frequency f of small oscillations of the bead about
the line joining the positive charges.
Solution:
The force of attraction of the bead (negative charge q) and each positive
charge +Q is of magnitude
Fo =
kQ |q|
kQ |q|
= 2
2
r
d + x2
Each force points toward the positive charge in question. The vertial component of the total force on the bead is zero, by “up-down symmetry”.
The net force on the bead is thus horizontal, and of magnitude 2Fo cos θ,
where θ is the angle between a force direction and the −x-direction. The
angle θ occurs in a √
right triangle with adjacent side x, opposite side d, and
hypoteneuse is r = d 2 + x2 . Thus,
x
cos θ = x/r = √
2
d + x2
When x is extremely small in comparison to d; that is, x ≪ d; we may drop
x in comparison to d in the denominator to obtain r ≈ d and cos θ ≈ x/d.
Thus the horizontal force on the bead is (approximately)
kQ |q| x 2kQ |q|
· =
x
d2
d
d3
The direction of the resultant force is to the left; that is, the −x direction.
According to Newton’s second law, ~F = m~a, the bead should accelerate to
the left when x is positive, with x-component
Fx = −2Fo cos θ ≈ −2 ·
2kQ |q|
x
md 3
Since the restoring force is opposite in direction to the displacement, this
corresponds to simple harmonic motion, as seen from the relation a = −ω2 x.
We therefore obtain
2kQ |q|
ω2 =
md 3
By putting in the numbers, we obtain
ax = Fx /m = −
Fo =
kQ |q| (9 × 109) · 10−6 · 10−6
≈
= 9 × 10−3 N
d2
(1)2
Thus
x
N
= −1.8 × 10−2
· x(mt)
1 mt
mt
Thus, “effective spring constant” is present; ke f f ≈ 1.8 × 10−2 N/mt. We
compute the corresponding angular velocity:
Fx ≈ −2 · (9 × 10−3 N)
ω2
ω
f = ω/(2π)
T = 1/ f
=
≈
≈
≈
ke f f /m ≈ 18
4.2 rad/sec
0.7 Hz
1.4 sec
Could this vibration be used to create an electric clock? Why or why not?
9. PHYS 221 - 005 QUIZ 3 28 September 2006
A ring of radius R contains a total charge q, distributed uniformly along it.
A second ring, placed concentrically and in same plane as the first ring, is of
radius 2R, and it contains total charge Q, again distributed uniformly along
it. The electric field along the axis of the ring at a distance R from its center
is measured and found to be zero. Determine the charge Q on the second
ring. You may express your answer in terms of q and R.
Solution:
We begin by considering the electric field produced by the inner ring, which
contains a charge q spread uniformly over its circumference, a circle of
radius r, along the axis of symmetry and a distance z from the center of the
circle. The electric field E1 lies along the symmetry axis. We may calculate
that component E1z by integrating over the charge distribution:
E1z =
The distance r =
quence
√
Z
dE1z = k
Z
dq
cos θ
r
√
R2 + z2 , and cos θ = z/r = z/ R2 + z2 . As a conse-
E1z = k
Z
√
dq
R2 + z 2
√
z
R2 + z 2
=
kqz
(R2 + z2 )3/2
The field from the second ring is obtained by making the replacements q →
Q and R → 2R, to obtain
E2z ==
kQz
(4R2 + z2 )3/2
The total field on axis is
Ez = E1z + E2z =
kqz
kQz
+
(R2 + z2 )3/2 (4R2 + z2 )3/2
That field vanishes at the point z = R, under the condition
kQR
kqR
√
+ √
=0
2 2R3 5 5R3
or
√
5 5
Q=− √ q
2 2
The charges on the ring must be of opposite signs for the field to vanish at
the point in question.
10. PHYS 221 - 006 QUIZ 3 29 September 2006
Three particles, each with positive charge Q, form an equilateral triangle,
with the sides of length a. What is the magnitude and direction of the electric field produced by the particles at the midpoint of a particular side?
Solution:
Let us draw the equilateral triangle with a a horizontal base, and take the
point in question on the base. There are two charges on that base, at distances a/2 from the that point. Their fields are each of magnitude kQ/(a/2)2,
and lie in opposite directions,
cancel. In addition, there is a
p so that they √
charge Q at a height h = a2 − (a/2)2 = 3a/2 above the point, which
produces a downward electric field
Ez −
kQ
4kQ
=− 2
2
h
3a
Thus, the net electric field is downward, with that downward component.
11. PHYS 221 - 003 QUIZ 3 05 October 2006
A thin (insulating) disk of radius R contains a total charge Q imbedded
uniformly into its surface.
• Determine the magnitude and the direction of the electric field at points
along the axis of the disk, at distance z from its axis.
• A second identical disk contains total charge −Q that is imbedded
uniformly into its surface. It is placed coaxially with the first disc,
with the two disks lying in parallel planes. The separation of these
two discs is 2R.
• Determine the magnitude and direction of the electric field at a point
halfway between the disks, along the axis of symmetry.
Solution:
Let us begin by calculating the electric field on axis at a distance z from
the first disk. It lies along that axis, by symmetry. Let us divide the disk
into thin concentric “onion rings” of radius r, which contain a total charge
dq. Since the charge density on the disk is σ = Q/(πR2), the charge in a
ring is dq = σ · dS. where dS is area of a ring of radius r and thickness dr;
dS = 2πrdr. Thus, dq = 2Qrdr/R2 , and the axial component of the electric
field produced by hat ring is
dEz = dE cos θ =
kdq
r 2 + z2
√
z
r 2 + z2
=
kzdq
(r2 + z2 )3/2
=
rdr
2kQz
2
2
R (r + z2 )3/2
The total electric field may be found by integrating over r:
Z
Z
2kQz R
rdr
1
2kQz R
= 2
d −√
Ez =
R2 0 (r2 + z2 )3/2
R
0
r 2 + z2
2kQz 1
1
2kQ
z
−√
=
= 2 1− √
2 + z2
R2
z
R
R
R2 + z 2
σ
z
=
1− √
2
2ε0
R + z2
We set z = R to obtain the electric field from the first disk:
σ
1
E1z =
1− √
2ε0
2
The second disk is negatively charged, and the same distance from the point
in question, so that the total field from the two disks is twice this value:
σ
1
Ez =
1− √
ε0
2
12. PHYS 221 - 004 QUIZ 3 10 October 2006
A point charge +Q is placed inside a thin hollow isolated conducting spherical shell of radius R — at the center of that shell, as shown. A total charge
−2Q is placed on that conducting shell, as well. Determine the magnitude
and direction of the electric field everywhere inside and outside the shell.
Solution:
Let us apply Gauss’s Law to an arbitrary concentric sphere of radius r:
ε0
I
~ = Qenc
~E · dS
Because of spherical symmetry, the electric field on that spherical surface
has only a radial field Er , which is independent of location on the surface.
Thus
ε 0 Er
I
dS = Qenc
ε0 Er 4πr2 = Qenc
kQenc
Qenc
= 2
Er =
2
4πε0 r
r
The only charge inside the conducting shell (r < R) is a point charge +Q at
the center, so that the electric field is radially outward and of magnitude
Er =
kQ
r2
Eithin the conductor, the electric field must vanish. Consequently, there
must be net charge −Q distributed uniformly ovr the inner surface of the
conductor. In addition, a charge −Q must be distributed uniformly over the
outer surface of the conductor. For r > R the net charge inside the gaussian
surface is Q − 2Q = −Q. Thus the electric field there is radially inward:
Er = −
kQ
r2
13. PHYS 221 - 005 Quiz 4 12 October 2006
Two identical solid, insulating spheres of radius R contain uniformly distributed charge throughout their interiors. The sphere on the left contains
total charge Q1 , whereas the right sphere contains total charge Q2 . The
spheres are placed tangentially at the surface, with their edges touching at
a point. Draw the line joining the centers of the spheres. We observe that
the electric field vanishes along that line, halfway from the center of the left
sphere, and toward the right sphere. Determine the ratio Q1 /Q2 .
Solution:
The location in question lies inside the left sphere, at a distance r = R/2
from its center, as well as outside the right sphere, at a distance r = 3R/2
from its center. Accordingly, the electric field E1 arising from the charge in
the left sphere it directed toward the right. Its magnitude can be computed
from Gauss/s Law:
E1 =
kQ1 r kQ1
kQenc kQ1 r3
= 2 3= 2 = 2
2
r
r R
R
2R
The electric field E2 arising from the charge in the right sphere is directed
to the left, and its magnitude is given by
E2 =
kQenc kQ2 4kQ2
= 2 =
r2
r
9R2
We thus have
E1 = E2
kQ1
4kQ2
=
2
2R
9R2
8
Q1 =
Q2
9
Note that the charges on the two spheres are of the same sign, and nearly
equal in magnitude.
14. PHYS 221 - 006 QUIZ 4 13 October 2006
A charged particle of charge Q is attached in place at the center of a thick
conducting spherical shell. There is no net charge upon the shell itself. The
inner radius of the shell is a, and its outer radius is b, as shown.
Determine the magnitude and direction of the electric field in each of these
three regions:
• r < a: inside the shell
• a < r < b: within the shell
• r > b outside the shell
Solution:
This is a problem involving spherical symmetry. Accordingly, we choose
Gaussian surfaces that are spheres, centered at the center of the shell. The
electric field for always lies in a radial direction; let its radial component be
Er .For any such Gaussian surface, we apply Gauss’s law to obtain:
ε0
I
~ = Qenclosed
~E · dS
ε0 Er (4πr2 ) = Qenclosed
Qenclosed
k Qenclosed
Er =
=
2
4πε0 r
r2
For r < a, the only charge inside the Gaussian surface (sphere or radius r)
is the point charge at the center, so that Qenclosed = Q, and
Er =
kQ
r2
For a < r < b, the surface of the sphere lies entirely within the conductor,
and the (static) electric field is zero there. Thus Er = 0 and Qenclosed = 0.
There must be a total charge −Q on the inside surface of the shell, at r = a.
For r > b the total charge enclosed in the sphere of radius r is Q, since the
shell itself is electrically neutral. For this region we obtain
Er =
kQ
r2
There must be a net charge +Q on the outer surface of the conducting shell,
to maintain its overall neutrality.
15. PHYS 221 - 004 QUIZ 4 24 October 2006
A 1.0 nanoFarad (nF) capacitor is charged to a potential difference of 20
Volts, and the charging battery is then disconnected.
This 1.0 nF capacitor is THEN connected in parallel with a second (initially uncharged) capacitor of unknown capacitance C. When equilibrium
is reached, the potential difference across the plates of the 1.0 nF capacitor
is then measured to be 5.0 Volts.
• What is the value of the unknown capacitance C, in nanoFarads?
• How much electrical energy (in Joules) is stored in the 1.0 nF capacitor just after the battery is disconnected?
• How much electrical energy is stored in each of the capacitors when
equilibrium is reached?
Solution:
The charge that is initially on the 1 nF capacitor is
Q0 = CV = 1.0 × 10−9 · 20 = 2.0 × 10−8C
The electrical energy initially stored in the capacitor is
1 2 1 −9
CV = (10 )(20)2 = 200 nJ
2
2
After the connection, that capacitor has a potential of 20 V between the
plates, and its final charge is
Q1 = 1.0 × 10−9 · 5 = 0.5 × 10−8C
The charge on the second capacitor is thus Q2 = Q0 − Q1 = 1.5 × 10−8C.
Since that second capacitor also has a potential of 5 V between its places,
Its capacitance is
C2 = Q2 /V2 = 1.5 × 10−8/5 = 3 nF
The energies stored in the capacitors at equilibrium are
1
C1V12 =
2
1
= C2V22 =
2
E1 =
E2
1 −9
(10 )(5)2 = 12.5 nJ
2
1
(3 × 10−9 )(5)2 = 37.5 nJ
2
The total electrical energy at equilibrium is thus 50nJ. Consequently, 200nJ
of electrical energy has been dissipated as heat in this process.
16. PHYS 221 - 003 QUIZ 4 26 October 2006
A capacitor with unknown capacitance C and initial potential difference
40 Volts is discharged through a resistor of unknown resistance R, when a
switch between them is closed at time t = 0 sec. At time t = 5 sec, the
potential difference across the resistor is measured to be 15 Volts.
• What is the time constant of this RC circuit, in seconds?
• What is the potential difference across the capacitor at time t = 15 sec?
Solution:
The charge on the capacitor at time t is
Q(t) = Q0 e−t/τ
where τ is the time constant for the circuit. Since Q = CV , we may express
the potential across the capacitor at time t in terms of the potential at t = 0
and the time constant τ:
V (t) = V0 e−t/τ
15 = 40 e−5/τ
0.375 = e−5/τ
5
−0.98 = −
τ
5
τ =
= 5.1 sec
.98
The potential difference at t = 15 sec is
V (15) = 40 · e−15/5.1 = 40 · 0.053 = 2.1 V
One may also write this result as
V (15) = 40 · (15/40)3 = 2.1 V
17. PHYS 221 - 005 QUIZ 5 02 November 2006
In a capacitor of capacitance 800 nanoFarads, the charge gradually “leaks”
from one plate to the other over the course of time. Suppose that the capacitor is initially charged to a potential of 30 Volts. After 24 hours, the potential
across its plates has dropped to 20 Volts. By considering an appropriate RC
circuit, determine the “equivalent resistance” between the plates, in Ohms.
Solution:
The charge on the capacitor at time t is given in terms of its charge Q0 at
time t = 0 and the time constant t = RC as
Q(t) = Q0 e−t/RC
Since Q(t) = CV (t), where C is the capacitance, it follows that
V (t) = V0 e−t/RC
30
= 1.5 = et/RC
20
24 · 3600 sec
ln 1.5 ≈ 0.405 =
RC
4
8.64 × 10
≈ 2.66 × 1011 Ω
R=
−7
0.405 · (8 × 10 )
One may obtain an approximate solution by saying that, in 24 hours, the
charge leaked across the capacitor is
∆Q = C∆V = (8 × 10−7 ) · 10 = 8 × 10−6 C
Thus, the average current flowing during this time is
I=
∆Q
8 × 10−6
=
= 9.2 × 10−11 A
∆t
8.64 × 104
The resistance R is given by the average Voltage across the capacitor (25
Volts) divided by this current:
R=
25
= 2.7 × 1011 Ω
9.2 × 10−11
This resistance is high, because the capacitor remains charged for a long
time.
18. PHYS 221 - 006 QUIZ 5 03 November 2006
Two wires, A and B, are made from different materials, with different crosssectional areas. The wires are each cut to a length of 1 meter, an connected
in series. A current of 3.0A passes through the wires.
Wire
A
B
Resistivity
Area
−6
1.0 × 10 Ωm 3.0 × 10−4 m2
2.0 × 10−6 Ωm 4.0 × 10−4 m2
What is the potential drop (in Volts) across each wire? At what rate is
electrical energy converted into heat (in Watts) in each wire?
Solution:
First, calculate the resistances of each wire:
ρA LA 1.0 × 10−6 1
=
= 0.0033 Ω
SA
3.0 × 10−4
ρB LB 2.0 × 10−6 1
RB =
=
= 0.005 Ω
SB
4.0 × 10−4
RA =
Thus the voltages across each wire are VA = RA I = 0.01 V and VB = RB I =
0.015 V .
The power lost in each resistor can then be computed:
PA = IVA = 0.03 W · · · PB = IVB = 0.045 W
19. PHYS 221 - 004 QUIZ 5 07 November 2006
A temperature-stable resistor is made by connecting a resistor made of Silicon in series with one made of Iron. If the required total resistance is 1000
Ohms, at temperatures around 20oC, what should the respective resistances
of the Silicon and the Iron resistor be?
Note: Temperature coefficients of resistivity a (in K −1 ):
• Iron: 6.5 × 10−3
• Silicon: −70 × 10−3
Solution:
At the reference temperature of 20oC, the resistances must add to 1000 Ω:
RFe + RSi = 1000 Ω
When the temperature changes by an amount ∆T , the individual resistances
change, but the total resistance remains the same:
RFe(1 + αFe ∆T ) + RSi (1 + αSi ∆T ) = 1000 Ω
As a consequence
αFe RFe + αSi RSi = 0
6.5 RFe − 70 RSi = 0
RFe = 10.8 RSi
Thus we obtain
1000 Ω
= 85 Ω
11.8
= 1000 Ω − RSi = 915 Ω
RSi =
RFe
Notice that, because the (negative) temperature coefficient of resistivity of
the semiconducting material silicon is much larger in magnitude than the
(positive) temperature coefficient for iron, the resistance of the silicon resistor is much less than that of the iron resistor.
20. PHYS 221 - 003 QUIZ 5 09 November 2006
A flash Lamp L is placed across a capacitor of capacitance C = 0.20 µF.
The combination is connected in series with a resistor of resistance R =
5 × 104 Ω and a battery of Voltage 60 Volts. The switch is closed at time
t = 0. How much time elapses before the Voltage across the lamp reaches
40 Volts - at which point the lamp flashes briefly?
Note: The flash lamp does not conduct electricity at all until the Voltage
across it reaches 40 Volts.
Solution:
The time constant for the circuit is
τ = RC = (2.0 × 10−7 F) · (5 × 104 Ω) = 0.01 sec
The charge on the capacitor is obtained from Kirchhoff’s loop equation to
be
Q(t) = CE (1 − e−t/τ )
Thus the Voltage across the capacitor is
V (t) = Q(t)/C = E (1 − e−t/τ )
The battery voltage is E = 60 V , and the time at which the Voltage across
the capacitor is 40 V satisfies the relation
40 = 60(1 − e−t/(.01 )
1/3 = e−t/(.01)
3 = et/τ
ln 3 ≈ 1.10 = t/(.01)
t ≈ 1.1 × 10−2 sec
This circuit produces over 80 flashes per second.
21. PHYS 221 - 005 QUIZ 6 16 November 2006
The Dees of a cyclotron of radius 80 cm are operated at an oscillator frequency of 6.0 MHz to accelerate protons.
• What is the (uniform) magnetic field in the cyclotron?
• What is the speed of the protons that leave the cyclotron, in meters per
second?
• What is the kinetic energy of the protons that leave the cyclotron, in
electron Volts?
Note:
• e0 = 1.6 × 10−19 C
• m p = 1.67 × 10−27 kg
• m p c2 = 938 MeV
Solution:
The force produced by the magnetic field must equal the mass of the proton
multiplied by its (centripetal) acceleration:
mv2
= qvB
r
qB
v
=
= 2π f
ω=
r
m
Thus
B=
and
m(2π f ) (1.67 × 10−27) · (2π) · (6 × 106))
=
= 0.39 T
q
1.6 × 10−19
v = ωR = 2π(6 × 106 ) · (0.8) = 3.0 × 107 m/s
The kinetic energy is
1 2 1
mv = (1.67 × 10−27 ) · (3.0 × 107)2 = 0.76 × 10−12 J
2
2
−13
7.6 × 10
=
eV = 4.7 MeV
1.6 × 10−19
K =
The speed is about ten percent of the speed of light, and the non-relativistic
approximation is fairly good.
22. PHYS 221 - 006 QUIZ 6 17 November 2006
The current density inside a long, solid, cylindrical wire of radius R lies
in the direction of the central axis. Its magnitude varies linearly with the
distance r from that central axis, J = J0 r/R.
• Calculate the total current I flowing in the wire, expressed in terms of
J0 and R.
• Find the magnitude and direction of the magnetic field everywhere
inside the wire.
Solution:
The total current passing through a circle of radius r that is concentric with
the axis of the wire may be computed by integration of the current density:
Ienc (r) =
Z
JdA =
Z r
0
r
2πJ0 r3
J0 2πrdr =
R
R 3
Thus the total current inside the wire is Ienc (R) = 2πJ0 R2 /3.
The magnetic field inside the wire is given by Ampére’s Law:
I
~ = µ0 Ienc
~B · dℓ
2πµ0 J0 r3
R 3
µ0 J0 r2
=
3R
Bt · (2πr) =
Bt
23. PHYS 221 - 004 QUIZ 6 28 November 2006
A solenoid having an inductance of L = 2 mH is connected in series with a
R = 40 kΩ resistor and an E = 30 Volt (DC) battery. The switch is closed
at time t = 0.
• What is the final current flowing in the circuit a long time after the
switch is closed? How much energy is stored in the inductor at that
time?
• At what time after the switch is closed is the current equal to 95 percent
of its final value?
Solution:
According to Kirchhoff’s Loop Equation,
E =L
dI
+ RI
dt
At very long times, the current is no longer changing, so that E = rI, so that
I∞ =
E
R
= 304 × 104 = 7.5 × 10−4 A
The energy stored in the inductor is
1
1
U = LI∞2 = · 2 × 10−3 · (7.5 × 10−4)2 = 5.6 × 10−10 J
2
2
From Kirchhoff’s Loop Equation, and the requirement that at t = 0 the current vanishes, we obtain
dI R
+ I
L
dt L
d h Rt/L i
E d h Rt/L i
e
=
Ie
R dt
dt
E h Rt/L i
e
− 1 = I(t)eRt/L − I(0)
R
i
Eh
I(t) =
1 − e−Rt/L
R
E
=
When the current equals 0.95I∞, we have
0.95 = 1 − e−Rt/L
e−Rt/L = 0.05
Rt
= 3.0
L
3L 3 · (2 × 10−3)
t =
=
= 1.5 × 10−7 sec
−4
R
4 × 10
24. PHYS 221 - 003 QUIZ 6 30 November 2006
An oscillating LC circuit consists of a C = 3.0 m f (microFarad) capacitor
and an L = 20 mH (milliHenry) inductor. The maximum voltage across the
capacitor is Vm = 40 Volts.
• What is the maximum charge on the capacitor?
• What is the maximum current through the inductor?
• What is the frequency of oscillation of the charge on the capacitor?
Solution:
The maximum charge on the capacitor is expressed in terms of the capacitance C and the maximum voltage Vm as
Qm = CVm = (3 × 10−6 ) · 40 = 1.2 × 10−4 C
The maximum energy in the capacitor Q2m /(2C), is equal to the maximum
energy in the inductor LIm2 /2:
1 2
1 Q2m
=
LI
2 C
2 m
1
1.44 × 10−8
Im2 = Q2m
=
= 0.24
LC
6 × 10−8
Im = 0.49A
The resonant angular frequency is
1
1
1
ω= √
=√
=
= 4100 rad/sec
LC
6 × 10−8 2.45 × 10−4
Then f = ω/(2π) = 600 Hz
Fall 2006 Examinations:
PHYS 221 - 003/004, TEST 1, 27 September 2006
1. [20 points] A ideal massless spring with spring constant k = 20 N/m hangs
vertically from the ceiling. A body of mass m = 0.5 kg is attached to its
free end and then released. Assume that the spring was not stretched at all
before the release.
• How far below its initial position x = 0 does the body descend before
turning around?
• What is the amplitude and period of the resulting simple harmonic
motion?
Solution:
The mechanical energy of this system at a displacement x below the starting
point, when the body has speed v, is
1 2
1
kx − mgx + mv2 = 0
2
2
This quantity vanishes, because body is released with speed v = 0 at position
x = 0. The body stops at position
x=
2mg 2 · 0.5 · 9.8
=
= 0.49 m
k
20
The block executes SHM about the position x = mg/k = 0.245 m, with amplitude A = 0.245 m. This can be seen by writing the relation of energy
conservation in the form
1 mg 2 1 2 1 mg 2
k x−
+ mv = k
2
k
2
2
k
The period of this motion is
r
r
m
0.5
2π
T = 2π
= 2π
= √ ≈ 1sec
k
20
40
Note that x = mg/k is the equilibrium position of the body when suspended
by the spring.
2. [20 points] An organ pipe produces sound in two normal modes corresponding to adjacent resonant frequencies of exactly 500 Hz and 600 Hz, respectively. The velocity of sound in air is 350 meters per second.
• Are both ends of the pipe open, or is one end closed?
• What is the lowest resonant frequency (fundamental mode) of this organ pipe?
• How long is the pipe?
Solution:
Let us first consider the case in which both ends of the pipe are open. The
ends of the pipes must be anti-nodes in that case, so that precisely n halfwavelengths of sound lie inside the pipe: L = nλ/2, or λ = 2L/n. The
resonant frequencies are
c
nc
fn = =
λ 2L
Thus, adjacent frequencies fn and fn+1 satisfy the relation
n+1
fn+1
=
fn
n
1
600
= 1+
500
n
1
1 + 0.2 = 1 +
n
n = 5
In this case, the fundamental frequency is f1 = 100 Hz, and the length of
the pipe L satisfies the relation
c
350
=
= f1 = 100
2L
2L
Consequently L = 1.75 m.
For the case of open and closed ends, the closed end is a node and the open
end is an anti-node, so that L = (2n + 1)λ/4. Consequently,
fn =
(2n + 1)c
4L
and
2n + 3
fn+1
=
fn
2n + 1
600
2
= 1+
500
2n + 1
2
1 + 0.2 = 1 +
2n + 1
n = 4.5
This case is impossible, since n is not an integer, as required.
3. [20 points] A uniform string of mass 30 grams and length 2.0 meters with
fixed ends is driven in its fundamental mode, so that the amplitude of motion
of the center of the string is 0.5 cm. The tension in the spring is 80 Newtons.
Note: Neglect gravity.
• Determine the resonant frequency of this fundamental mode.
• Determine the maximum velocity of a point at the center of the string.
• Determine the maximum acceleration of a point at the center of the
string.
Solution:
The wavelength of vibrations of the string in its fundamental mode is λ =
2L = 4.0 m. Thus the mass per unit length of the string is µ = m/L =
0.03/2.0 kg/m = 0.015 kg/m. The velocity of transverse vibrations is
s
T √
v=
= 800.015 ≈ 72 m/s
µ
The resonant frequency is thus
f=
v 72
=
= 18 Hz
λ
4
The corresponding angular frequency is ω = 2π f ≈ 110 rad/sec. The displacement of the string at position x; (0 < x < L) and time t is
y(x,t) = A sin
πx
cos(ωt)
L
At x = L/2, we obtain
π
y(t) = A sin cos(ωt) = A cos(ωt)
2
Thus
dy
= −ωA sin(ωt)
dt
dvy
ay =
= −ω2 A cos(ωt)
dt
vy =
The maximum velocity is vm = ωA = (110) · (0.005) = 0.55m/s, whereas
the maximum acceleration is am = ω2 A = (0.55) · 110 = 65m/sec2 . This
maximum accelation at the center is more than 6 g’s.
4. [20 points] Two tiny conducting balls of identical mass (m = 1 gram) and
identical charge (Q = 1 microCoulomb = 10−6C) hang from non-conducting
threads of length L, which are attached to a point on the ceiling. The threads
each make an angle of θ = 30o with the vertical axis.
• Show all the forces acting on the balls, including their weight.
• Determine the length of the thread, in meters.
Solution:
The point on the ceiling and the two balls lie at vertices of an equilateral
triangle, so that the distance between the balls is L. Consequently, the force
of Coulombic repulsion between the ball is F = kQ2 /L2 . The three forces
acting on the ball – its weight mg (downward), the tension in the string (30o
from the vertical, and the Coulombic repulsion (horizontal) – must sum to
zero. Consequently, these vectors lie along a 30o − 60o − 90o triangle, so
that
1
F
tan 30o = √ =
mg
3
2
mg 0.001 · 9.8
kQ
= 0.0057 N
F= 2 = √ =
L
0.577
3
kQ2
9 × 10−3
√ =
L2 =
= 1.5
0.0057
mg/ 3
L = 1.2 m
This is a point of stable equilibrium.
5. [20 points] A total charge Q is distributed uniformly around a thin circular
ring of radius R.
• Determine the magnitude and direction of the electric field along the
axis of symmetry of the ring, a distance z from the center of the ring.
• Determine the distance at which the magnitude of the electric field is
a maximum. You may express the answer in terms of R.
Solution:
By symmetry, the electric field lies along the axis of symmetry of the ring.
The contribution of an infinitesimal piece of charge on the ring to that field
component is
kdq
dEz = 2 cos θ
r
where r2 = R2 + z2 and cos θ = z/r. The total electric field may be obtained
by integration over the charge distribution of the ring:
Ez =
Z
kz
dEz = 2 2 3/2
(R + z )
Z
dq =
kQz
(R2 + z2 )3/2
To obtain the value of z corresponding to the maximum field, we calculate
dEz /dz and set it to zero:
kQ
dEz
2
2
2
(z
+
R
)
−
3z
=0
= 2
dz
(z + R2 )5/2
√
We get R2 − 2z2 = 0, or z = R/ 2.
6. [Extra Credit; 10 points] A particular 500 Hz isotropic sound source is
barely audible (at the threshold of hearing) at a distance of 2 kilometers.
Note: Please ignore background noise, sound absorption, sound reflection,
etc.
• How much acoustic power is being produced at the source?
• At what distance from the source would the intensity be at the 50 dB
(deciBel) level, corresponding to normal conversation.
Solution:
The intensity at the threshold of hearing is Ith = 10−12 W /m2 . At a distance
r = 2.0 × 103 m, the acoustic power is given by
P = Ith × (4πr2 ) = 4π(2.0 × 103)2 10−12 = 16π × 10−6 W = 5 × 10−5 W
At β = 50 dB, the intensity satisfies the relation
β = 50 = 10 log10
105 =
I
Ith
I = 10−7 W /m2 =
I
Ith
P
4πr2
Thus
r2 =
and r = 6.5 m.
P
5 × 10−5
=
= 40
4πI 4π × 10−7
PHYS 221 - 005/006, TEST 1, 27 September 2006
1. [20 points] A 200 gram stone is attached to the bottom of an ideal massless
spring that is suspended from the ceiling. The stone is pulled downward
from its equilibrium position, and then released (vertical motion). If the
maximum speed of the stone is 50 cm/sec, and the period is 1.0 sec, find
the following:
• The spring constant k.
• The amplitude of motion (relative to the equilibrium position).
• The frequency of oscillation.
Solution:
The frequency of oscillation of the stone is f = 1/T = 1 Hz. The angular
frequency is ω = 2π f = 6.3 rad/sec. The corresponding spring constant is
k = mω2 = 0.2 · (6.3)2 = 8 N/m The maximum speed is vm = ωA, so that
the corresponding amplitude is A = 0.5/(6.3) = 0.08 m.
2. [20 points] Two uniform strings, each of length 80 cm, are held at the same
tension of 100 Newtons. The first string has a mass 10.0 grams, whereas the
second has mass 10.5 grams. Each of the strings is driven in its fundamental
mode.
• Determine the frequency of vibration of the first string.
• Is the vibrational frequency of the second string higher or lower than
for the first spring? Explain.
• Determine the frequency of beats for the two strings.
Solution:
The wavelength of the fundamental mode of the first string is λ = 2L =
1.6 m. The mass per unit length of that string is µ = m/L = 0.01/0.8 =
0.012 kg/m. The velocity of transverse vibrations of that string is
v=
s
T √
= 100 · 80 = 89 m/s
µ
The corresponding frequency of vibration is f = v/λ = 56 Hz.
The frequency of a vibrating spring in the fundamental mode is given by
1
f=
2L
s
T
µ
As a consequence for the two strings the ratio of frequencies is
f
=
f′
s
µ′
=
µ
r
10.5
= 1.025
10
Thus, the frequency f ′ for the second spring is less than f for the first spring.
We have
f − f ′ = .025 f = 1.4 Hz
3. [20 points] A violin string of mass 1.0 grams and length 20 cm produces
sound at a fundamental frequency of 500 Hz.
• Determine the tension in the string.
• Determine the wavelength and frequency of transverse waves on the
string.
• Determine the wavelength of the sound produced in the air.
Note: The velocity of sound in the air is 350 meters per second.
Solution:
The mass per unit length of the violin string is µ = m/L = 0.001/0.2 =
0.005 kg/m. The wavelength for the fundamental mode is λ = 2L = 0.4 m.
The velocity of transverse vibrations is
v = λ f = 200 m/s =
s
T
µ
The tension is thus given by
T = (0.005) · (200)2 = 200N
The frequency of sound in air is also 500 H. The wavelength of sound is
given by
λ′ = vsound / f = 350/500 = 0.7 m
4. [20 points] Four charges, +Q, +2Q, −Q, and −2Q, are placed at vertices
of a square of side a, at these respective locations: (0, 0), (a, 0), (a, a), (0, a).
Determine the magnitude and direction of the net electrostatic force acting
on the charge +Q, which is located at (0, 0).
Solution:
The forces caused by the other three charge are tabulated below:
charge f orce
2kQ2
+2Q
a2
−Q
−2Q
kQ2
2a2
2kQ2
a2
total
x − comp
2
− 2kQ
a2
kQ2
√
2 2a2
0
kQ2
a2
1
( 2√
− 2)
2
y − comp
0
kQ2
√
2 2a2
2kQ2
a2
kQ2 √
( 1 + 2)
a2 2 2
Thus,
~F = kQ −1.65î + 2.35 ĵ
2
a
The magnitude of the force is 2.87kQ/a2, and its direction is 124o from the
+x-axis.
5. [20 points] An insulating wire of length L has a total charge Q deposited
uniformly along its length. Determine magnitude and direction of the electric field at a point along the (extended) line of the wire, a distance x beyond
the end of the wire.
Solution:
Let the wire lie in a horizontal line, 0 ≤ x ≤ L, and let the point P be a
distance y to the right if that line. The electric field points to the left. Its
magnitude is given by
Ey =
Z
kdq
r2
The charge dq in a piece of the wire oflength dx is given by dq = Qdx/L.
If that charge lies at location x, then its distance to the point P is r = x + y.
Thus
Z L
kQ
dx
2
0 L (x + y)
1 x=L
kQ
=
−
L
x + y x=0
kQ
1
1
=
−
+
L
L+y y
kQ
L
kQ
=
=
L y(y + L) y(y + L)
Ey =
6. [Extra Credit; 10 points] One can determine the speed of blood flowing in
an artery in the body by measuring the frequency shift of reflected high
frequency ultrasound. Sound is produced at a frequency of 5.0 MHz, and
reflected signal in blood has its frequency increased by about 5.0 kHz. Determine the speed of flow of blood in that artery.
Note: Assume that the blood is flowing directly toward the sound source,
and that the velocity of sound in blood is about 1500 meters per second.
Solution:
The incident wave has frequency f0 = 5.0 MHz. The frequency of sound
reflected off the blood is f1 = f0 (1 + v/c), where c is the speed of sound in
blood. The speed of sound detected back at the source is
f2 = f1 /(1 − v/c) ≈ f1 (1 + v/c) ≈ f0 (1 + 2v/c)
The fractional change of sound is
∆ f / f = ( f2 − f0 )/ f0 ≈ 2v/c
The measured fractional change in frequency is ∆ f / f = 5/5000 = 0.001.
Thus v = c/2000 = 0.75 m/s.
PHYS 221 - 003/004, Test 2, 01 November 2006
1. [20 points] A positive charge of +3.00 nC (1nC = 10−9 C) is spread uniformly throughout the volume of a sphere of radius R = 0.5 meters. What is
the magnitude and direction of the electric field at the following distances r
from the center of the sphere?
• r = 2.0 meters
• r = 0.5 meters
• r = 0.1 meters
Solution:
The electrical field is radial, because of spherical symmetry. Take the Gaussian surface to be a concentric sphere of radius r. According to Gauss’s
Law
ε0
I
~ = Qenc
~E · dS
ε0 Er (4πr2 ) = Qenc
Qenc 1
Er =
4πε0 r2
kQenc
Er =
r2
Thus, at r = 2.0 m,
Er = (9 × 109 ) (3 × 10−9 )/22 = 6.75 N/C
In addition, at r = 0.5 m, Er = 108 N/C.
For r ≥ R, Qenc = Q and Er = kQ/r2 . Inside the sphere for r < R, Qenc =
Q(r/R)3 and
Er =
kQ r3
kQr
= 3
2
3
r R
R
Thus, for r = 0.1 m,
Er = (9 × 109 ) (3 × 10−9) (0.1)/(0.5)3 = 22 N/C
2. [20 points] An insulating sphere of radius R = 0.1 meters contains a charge
that is uniformly spread throughout its interior. It observed that there is a
net electric flux of 8 × 10−4 Nm2 /C passing out of a concentric spherical
Gaussian surface of radius r = 0.5 meters.
• What is the electric potential at a distance of 1.0 meters from the center
of the sphere, in Volts?
• What is the total charge on the sphere, in Coulombs?
Solution:
Gauss’s Law has the form ε0 ΦE = Qenc , so that the charge inside a concentric spherical Gaussian surface of radius 0.5 m is Qenc = (8.85 × 10−12 ) ·
(8 × 10−4) = 7 × 10−15 C. This is the charge inside the sphere. The electrostatic potential at an arbitrary point outside the sphere, at a distance r from
its center, is V = kQenc /r = (9 × 109) · (7 × 10−15)/1.0 = 6.4 × 10−5 V .
3. [20 points] A capacitor of unknown capacitance C is charged to 200 Volts,
and the charging source is then disconnected. Then it is connected across an
uncharged 50 µF capacitor. The final potential difference across the 50 µF
capacitor is 40 Volts. Note: 1 µF = 10−6 F.
• What is the unknown capacitance C?
• How much charge is on each capacitor at the conclusion?
• How much electrical energy is stored in the system, before and after
the connection? Does it change? Why or why not?
Solution:
Let q0 be the charge initially on the capacitor C; note that q0 = C(200).
That charge is distributed among the two capacitors after the connection:
q0 = q + q′ . Note that q′ = (50 × 10−6 ) (40) = 2.0 mC, and q = 40C. Thus
q0
200C
C
q0
=
=
=
=
q + q′
40C + 2 × 10−3
12.5 µF
200 · (12.5 × 10−6) = 2.5 mC
The energy initially stored in the capacitor is 1/2(12.5 × 10−6 )(200)2 =
250 mJ. The energy stored afterward is 1/2(62.5 × 10−6 )(40)2 = 50 mJ.
Note that 4/5 of the energy has been dissipated in the process.
4. [20 points] A steel trolley car rail has a cross-sectional area of 0.01 square
meters. The electrical resistivity of steel is 3.00 × 10−7 Ωm.
• What is the resistance of 10 km of trolley rail?
• If a current of 100 A is passing through the trolley rail from end to end,
how much energy per unit time is dissipated within the rail?.
Solution:
The resistance of the rail is
R=
ρL 3 × 10−7 · 104
=
= 0.3 Ω
A
10−2
The power dissipated is
P = I 2R = (100)2 (0.3) = 3000 W
5. [20 points] A capacitor of capacitance 2.0 µF “leaks” slightly, in that charge
passes from one plate to the other over the course of time. The charge on
one of the plates drops to half its value in 20 minutes.
Note 1 µF = 10−6 F.
What is the equivalent resistance between the capacitor plates?
Solution:
The charge on the capacitor at time t is
q(t) = q0 e−t/τ
Thus
q0 /2 = q0 e−τ
2 = et/τ
t
ln 2 = 0.69 =
τ
τ = 20 min/0.69 = 28.85 min = 1730 sec
The resistance R is
R=
1730
τ
=
= 8.7 × 108 Ω
C 2 × 10−6
6. [Extra credit; 10 points] A resistor of unknown resistance is connected between the terminals of an ideal 9.0 Volt battery. The rate of dissipation of
energy in the resistor is 0.05 Watts.
• Determine the resistance of the resistor.
• If the same resistor is placed between the terminals of an ideal 36 Volt
battery, determine the rate of dissipation of energy in the resistor, in
Watts.
Solution:
From the formula for power lost in the resistor P = V 2 /R, it follows that the
resistance is
92
V2
R=
=
= 1600 Ω
P
0.05
The power lost in the 36 V battery is
P′ =
362
= 0.80 W
1600
PHYS 221- 005/006, TEST 2, 01 November 2006
1. [20 points] Charge of uniform volume density ρ = +2.0 nC/m3 fills a large
(infinite) slab between x = −1.0 meters and x = +1.0 meters. There are no
other charges in the vicinity of this large slab. What is the magnitude and
direction of the electric field at any point with the following coordinates:
Note: 1nC = 10−9 C.
• x = −2.0 meters
• x = −0.2 meters
• x = +0.0 meters
• x = +0.6 meters
• x = +4.0 meters
Solution:
The charge distribution is symmetric about the plane x = 0, which lies at
the center of the slab. Thus, the electric field is zero along that slab, at
x = 0. The electric field outside lies in the direction of increasing x for
x > 0, whereas for x < 0 it lies in the direction of decreasing x. The charge
per nit area in the slab is σ = ρt, where T = 2 m is the thickness of the slab.
Thus, σ = 2 × 10−9 · 2 = 4 × 10−9 C/m2 . It follows from Gauss’s law for a
pillbox of area A parallel to the slab and enclosing a piece of it that that the
magnitude of the electric field E outside the slab is given by
ε0
I
~ = Qenc
~E · dS
ε0 E(2A) = σA
σ
4 × 10−9
=
= 230 N/C
E =
2ε0 2 · 8.85 × 10−12
To determine the field inside the slab at a distance x from its center, we
take a Gaussian pillbox of area A with bases at on its center plane and at a
distance x from the center. Applying Gauss’s law, we get
ε0
I
~ = Qenc
~E · dW
ε0 E(x)(A) = ρx
ρx
2 × 10−9 x
E =
=
= 230 x N/C
ε0
·8.85 × 10−12
Thus, at x = −0.2, Ex = −45 N/C, whereas at x = 0.6, Ex = +140 N/C.
2. [20 points] A charge +Q is distributed uniformly throughout a spherical
volume of radius R. Let the electrostatic potential V be zero at infinity.
• What is the electrostatic potential inside the sphere: r < R?
• What is the electrostatic potential outside the sphere: r > R?
• What is the potential difference between the center of the sphere (r =
0) and its surface (r = R)?
Solution:
The electric field is determined through Gauss’s Law with a concentric
spherical Gaussian surface. Outside the sphere (r > R) we obtain
ε0
I
~ = Qenc
~E · dS
ε0 Er (4πr2 ) = Q
dV
kQ
=
Er = −
dr
r2
kQ
V (r) =
r
Inside the sphere, Qenc = Qr3 /R3 , so that
kQ r3
kQr
= 3
2
3
r R
R
2
kQr
V (r) = − 3 +V0
2R
Er = −
dV
dr
=
The constant of integration is determined by the requirement that, at r = R,
V (R) = kQ/R, as the point is approached from both inside and outside the
sphere. Thus, V0 = 3KQ/(2R). Finally,
V (0) −V (R) =
3kQ kQ kQ
−
=
2R
R
2R
3. [20 points] A dielectric sphere capacitor is placed by putting dielectric material of dielectric constant κ = 50 between concentric spherical metallic
plates. The inner plate has a radius of r1 = 1.00 mm, and the outer plate has
a radius of r2 = 1.05 mm.
Note: 1 mm = 10−3 m.
• Determine the capacitance, in Farads.
• With a charge of 20 nC on the capacitor, how much electrical energy
is stored in it?
Solution:
Let us apply Gauss’s law to a concentric spherical surface lying entirely
within the dielectric:
ε0
I
~ = Qenc
κ~E · dS
ε0 Er · 4pir2 = −Q
kQ
Er = − 2
κr
The potential difference between the plates of the capacitor are
∆V = −
Z b
a
kQ
Er dr = −
κr
b
kQ 1 1
kQ b − a
=
−
=
= Q/C
κ a b
κ ab
a
The capacitance is
C=
κ ab
50 1.05 × 16−6
=
= 1.16 × 10−10 F
k b − a 9 × 109 5 × 10−5
The stored electrical energy is
U=
Q2
(2 × 10−8)2
=
= 1.7 × 10−6 J
−10
2C 2 · 1.16 × 10
4. [20 points] A fuse in an electric circuit is a wire that is designed to melt, and
thereby to open the circuit, if the current exceeds a predetermined value.
Suppose that the material to be used melts when the current density rises to
500 A/cm2 .
• What diameter of cylindrical wire should be used to make a fuse that
will limit the current to 1.0 A?
• What must be the resistivity ρ of the material in the wire, if there is an
electric field of 102 V /m in the wire when it melts?
Solution:
The current density in the wire is J = 5 × 106A/m2 , so that for an electric
field e = 100V /m in the wire, the resistivity must be ρ = E/J = 100/(5 ×
106 ) = 2 × 10−5 Ωm.
When a current I = 1 A is flowing, the area of the wire is
A = 1/(5 × 106) = 2 × 10−7 m2
Setting A = πD2 /4, we obtain a diameter D = 0.50 mm.
5. [20 points] Two ideal batteries, each of electromotive potential E = +12
Volts, are each connected in series with a 6 Ω resistors and then in parallel
with one other, with opposite polarities. This combination is then placed
across a third resistor, which also has a resistance of 6 Ω. Under steady-state
conditions, determine the magnitude and direction of the current through
each resistor, the power loss in each resistor, and the power provided by
each battery.
Solution:
The current passing through the third resistor is zero, since the batteries have
opposite polarities. Furthermore, the potential drop across each batteryresistor combination is zero. Thus, The current flowing through each resistor is I = 2 A. Each battery provides power P = IV = 24 W , and the resistors
each convert 24 W into thermal energy.
6. [Extra Credit; 10 points] In Millikan’s experiment, and oil drop of radius
2.0 µm and density 0.9 g/cm3 is suspended (at rest) in a chamber, with a
downward electric field of 2.0 × 105 N/C is applied. Find the charge on the
drop, in Coulombs. Do not neglect gravity.
Note: (1µm = 10−6 meters)
Solution:
For balance of the forces we obtain
qE = −mg
4πr3
g
qE = −ρ
3
4π r3 ρg
q = −
3 E
4π (2 × 10−6 )3 · 900 · 9.8
= −
3
2 × 105
= = 1.44 × 10−18 C
This corresponds to about nine electron charges.
PHYS 221 - 003/004, FINAL Examination, 13 December 2006
1. [25 points] The scale of a light spring balance that reads from 0 to m = 20 kg
is x = 8.0 cm long. A package suspended from the balance is found to
oscillate vertically from the spring balance with a frequency of 1.5 Hz.
• What is the spring constant k, in N/m?
• What is the mass of the package, in kg?
Solution:
The spring constant is k = mg/x = 20 · 9.8/0.08 = 2450 N/m. The angular
frequency of vibration is ω = 2π f = 2π(1.50) = 9.42 rad/sec. From the
relation ω2 = k/m we determine the mass:
m=
k
2450
=
= 28 kg
ω2 (9.42)2
2. [25 points] Two charged thin concentric spherical shells have radii 10 cm
and 20 cm, respectively. The charge on the inner shell is +6 nC, whereas
the charge on the outer shell is +12 nC. Find the electric field at these
distances from the center of the shells.
• r = 15cm
• r = 30 cm
Solution:
For a concentric spherical Gaussian surface of radius r, Gauss’s Law yields
ε0
I
~ = Qenc
~E · dS
ε0 (4πr2 )Er = Qenc
kQeenc
Er =
r2
For r = 15 cm, Qenc = 6 nC, so that
Er =
(9 × 109 ) · (6 × 10−9)
= 2400 N/C.
(0.15)2
For r = 30 cm, Qenc = 18 nC, so that
Er =
(9 × 109 ) · (18 × 10−9)
= 1800 N/C.
(0.30)2
3. [25 points] An air-filled parallel-plate capacitor has a capacitance of C0 =
1.5 pF. The separation between plates d is doubled, and a dielectric material is inserted between the plates. The new capacitance is C1 = 6.0 pF.
• Determine the dielectric constant κ of the material.
• If charges of ±3 microCoulombs (mC) remain on the plates of the
capacitor throughout this process, calculate the energy stored in the
capacitor (in Joules) in the beginning and in the end.
Solution:
The capacitance of the air-filled capacitor is C0 = ε0 A/d, whereas for the
dielectric-filled capacitor we obtain C1 = κε0A/(2d). It follows that
C1 /C0 = κ/2 = 4
Thus, the dielectric constant is κ = 8.
The energy originally stored in the capacitor is
Q20
(3 × 10−6)2
U0 =
=
=3J
2C0 2(1.5 × 10−12)
The energy stored at the end is
Q20
(3 × 10−6 )2
=
= 0.75 J
U1 =
2C1 2 · (6 × 10−12)
4. [25 points] In a certain cyclotron, protons move in a circle of radius 0.5
meters. The magnitude of the magnetic field is 3.0 T , and the direction of
the field is perpendicular to the orbital plane.
• What is the oscillator frequency?
• What is the speed of the protons, in meters per second?
• What is the kinetic energy of the protons, in electron Volts?
Solution:
It follows from the Lorentz force relation that
mv2
= qvb
r
v qB (1.6 × 10−19 ) · 3
=
=
= 2.87 × 108 rad/sec
ω =
−27
r
m
1.67 × 10
The corresponding frequency is f = ω/(2π) = 46 MHz. The speed of the
protons at ejection is v = ωR = (2.87 × 108) · (0.5) = 1.5 × 108 m/s. The
corresponding kinetic energy is
1
1
K = mv2 = (1.67 × 10−27 )(1.44 × 108)2 = 106 MeV
2
2
Although this is nearly half the speed of light, the non-relativistic approximation is fairly good.
5. [25 points] the inductance of a closely wound coil is such that an EMF of
3.0 mV is induced when the current changes at the rate of 5.0 Amps per
second. A steady current of 1.0 Amps through the coil produces a flux of
4.0 × 10−5 Webers through each turn.
• Calculate the inductance of the coil, in Henries.
• How many turns N are there in the coil?
Solution:
From Faraday’s Law,
E =−
dΦ
dI
= −L
dt
dt
it follows that the inductance is L = (3 × 10−3 )/5 = 6 × 10−4 H. The flux
per turn is Φ0 = BA = 4 × 10−5 W . Thus,
NΦ0 = Φ = LI
N · (4 × 10−5) = (6 × 10−4 )1
N = 15 turns
6. [25 points] As a parallel-plate capacitor with circular plates 30 cm in diameter is being charged, the current density of the displacement current between
the plates is uniform and has a magnitude of Jd = 20 A/m2 .
• Calculate the magnitude of the magnetic field B at a distance of 4 =
20cm from the center of symmetry of this region.
• Calculate dE/dt in this region.
Solution:
The displacement current passing through a concentric ring of radius r is
Id = πr2 Jd = π(0.2)2(20) − 1.41 A
According to Maxwell’s extension of Ampère’s Law
I
~ = µ0 Id
~B · dℓ
Bt (2πr) = µ0 Id
µo Id (4π × 10−7 ) · (1/41
=
= 1.41 × 10−6 T
Bt =
2πr
2π(−02)
From the relation
Id = ε0
Z
∂E
∂~E ~
· dS = ε0 A
∂t
∂t
We obtain
Jd = ε0
∂E
∂t
or
∂E Jd
20
V
=
=
= 2.3 × 10−12
−12
∂t
ε0 8.85 × 10
ms
7. [Extra credit; 10 points] You are standing at a distance D from an isotropic
point source of sound. You walk 100 meters directly toward the source, and
observe that the intensity of sound has doubled. Calculate the distance D.
Solution:
The intensity I of isotropically produced sound at a distance D from a source
of acoustic power P is I = P/(4πD2 ). At distance D − 100 the intensity is
2I, so that 2I = P/(4π(D − 100)2). Consequently
P = 4πD2 I = 8π(D − 100)2I
D2 = 2(D − 100)2
√
D =
2(D − 100)
√
100
√ = 100(2 + 2) = 340 m
D =
1 − 1/ 2
PHYS 221 - 005/006, Final Examination, 14 December 2006
1. [25 points] A nylon guitar string has a linear density of µ = 6.0 g/m and is
under a tension T = 120 N. The fixed supports are a distance D = 80 cm
apart. The string is oscillating in a standing wave pattern consisting of
three loops. Calculate the speed, wavelength, and frequency of the traveling waves whose superposition gives this standing wave.
Solution:
Each loop corresponds to a half-wavelength, so that D = 3/2λ, or λ = 2/3 ·
80 = 53 cm. The velocity of transverse vibrations is
v=
s
T
=
µ
r
√
120
= 200000 = 140 m/s
0.006
The transverse displacement y(x,t) is given by
y = A sin
3πx
A
sin ωt = [cos(kx − ωt) − kx + ωt)]
D
2
The frequency of vibration is f = v/λ = 140/0.53 = 265 Hz.
2. [25 points] Two particles, each with charges +12nC, are placed at two of
the vertices of an equilateral triangle with edge length a = 0.5 meters.
• What is the magnitude and direction of the electric field at the third
vertex of the triangle?
• What is the electrostatic potential at that point?
Solution:
The electric field at the third vertex is the sum of the fields produced by the
two charges: ~E = ~E1 + ~E2 . From symmetry, the field lies along the altitude
to the side joining the charges, and away from the triangle. The magnitude
of each of the fields is E1 = E2 = kQ/a2 . The magnitude of the total field is
√
E = 2E1 cos 30o = 3kQ/a2
√
3(9 × 109 ) · (12 × 10−9)/(0.5)2 = 720 V /m
=
The electric potential at the vertex is
V = V1 +V2 = 2kQ/a = 2(9 × 109 ) · (12 × 10−9)/(0.5) = 432 V
3. [25 points] An RC circuit is connected across a 20 Volt battery, and the
switch is closed at time t = 0. The resistance is R = 2000 Ohms, and the
capacitance is C = 80 microFarads.
• What is the final charge on the capacitor, in Coulombs?
• At what time after the switch is closed is the capacitor charged to half
of its final value?
Solution:
The equilibrium charge on the capacitor is Q0 = CV = 20 · 8 × 10−5 = 1.6 ×
10−3 C. The time constant for this circuit is τ = RC = 2 × 10−3 · 8 × 10−5 =
0.16 sec. The charge on the capacitor at time is Q(t):
h
i
−t/τ
Q(t) = Q0 1 − e
h
i
Q0
−t/τ
= Q0 1 − e
2
1
= e−t/τ
2
t
ln 2 =
τ
t = 0.69 · 0.16 = 0.11 sec
4. [25 points] A long solenoid with n = 5000 turns per meter carries a current
I. An electron moves within the solenoid in a circle of radius R = 3.0 cm,
perpendicular to the solenoid axis. The speed of the electron is v = 106
meters/second. Find the current in the solenoid.
Solution:
The Lorentz force relation ~F = m~a = q~V × ~B yields the relation
mv2
= qvB
r
mv
(9.1 × 10−31 ) · 106
B =
=
= 1.9 × 10−4 T
qR (1.6 × 10−19 ) · (0.03)
It follows from Ampère’s Law that
B = µ0 nI
B
1.9 × 10−4
I =
=
= 0.03 A
µ0 n (4π × 10−7) · (5 × 103)
5. [25 points] In an oscillating LC circuit, with L = 8.0 mH and C = 2.0 µF.
At time t = 0, the current is maximum at 20 mA.
• What is the maximum charge on the capacitor during the oscillations?
• At what earliest time t > 0 is the rate of change of energy in the capacitor a maximum? What is that maximum rate of change, in Joules
per second?
Solution:
At time t = 0 the charge on the capacitor is Q0 = 0, whereas the current in
the inductor is I0 = 0.02 A. According to Kirchhoff’s loop equation,
dI Q
+
= 0
dt C
1
d 2Q
+
Q = 0
2
dt
LC
L
The angular frequency of (harmonic) oscillation of charge is
Thus
1
1
ω= √
=p
= 7900 rad/sec
−5
LC
(8 × 10 )(2 × 10−8)
Q(t) = Q0 sin ωt
I(t) = ωQ0 cos ωt
It follows that the maximum charge on the capacitor is
Q0 = I0 ω = 0.02/7900 = 2.6 × 10−6 C
The energy in the capacitor at time t is U (t):
Q(t)2 Q20
=
sin2 ωt
2C
2C
ωQ20
ωQ20
U ′ (t) =
sin ωt cos ωt =
sin(2ωt)
C
2C
U (t) =
The maximum rate of change is
′
Umax
= ωQ20 /(2c) = (7900)(2.5 × 10−6)2 /(4 × 10−6 ) = 2.5 × 10−2 J
6. [25 points] The index of refraction of Benzene is n = 1.8.
• If a small coin lies at a depth of d = 15 cm in a pool of Benzene,
determine the apparent depth y of the coin, as seen from above.
• What is the critical angle for a light ray traveling in Benzene toward a
layer of air above the Benzene.
Solution:
If a light ray is incident from above at a small angle θ to the vertical direction, it is refracted to an angle φ to the vertical:
sin θ = n sin φ
tan θ ≈ n tan φ
x
x
≈ n
y
d
d
15
y ≈
=
= 8 cm
n 1.8
Note that the apparent horizontal location x of the coin is unaffected by
refraction.
The critical angle φ in Benzene is determined by the relation
n sin φ = sin 90o = 1
1
1
sin φ =
= = 0.55
n .8
φ = 33.7o
7. [Extra Credit; 10 points] How much work is required to assemble four
charges Q = 5 nC at the vertices of a square of side a = 0.2 meters? Assume
that the charges are infinitely far apart before this assembly.
Solution:
The potential energy stored in a configuration of charges qi is a sum of the
potential energies for each pair of charges:
qi q j
i< j ri j
U =k∑
Four of pairs of charges are
√ separated by a distance a, whereas the separation of other two pairs is 2a. Thus
√
Q2
kQ2
Q2
+ 2k √ =
(4 + 2)
a
a
2a
9
−9
√
(9 × 10 ) · (5 × 10 )2
=
(4 + 2) = 6.1 × 10−6 J
0.2
U = 4k